A066571 Number of sets of positive integers with arithmetic mean n.

1, 3, 9, 31, 117, 479, 2061, 9183, 42021, 196239, 931457, 4480531, 21793257, 107004891, 529656765, 2640160039, 13241371629, 66771501151, 338333343825, 1721768732423, 8796192611917, 45096680384635, 231945566136129, 1196461977291959, 6188390166782849

Offset

1,2

Comments

From Franklin T. Adams-Watters, Sep 13 2011: (Start)

If we use nonnegative integers instead of positive integers, we get this sequence shifted left (i.e., with offset 0).

The largest number that can be included in set of positive integers with mean n is the triangular number n*(n+1)/2 = A000217(n).

All values are odd. Sets including n are paired with the same set with n removed, with exception of {n}, as the empty set has no average.

(End)

Links

Martin Fuller, Table of n, a(n) for n = 1..200

Formula

Sum of coefficient of t^k x^(n*k) in Product_{i=1..n*k} (1+t*x^i) for k <= 2*n-1. - N. J. A. Sloane

From Martin Fuller, Sep 14 2023: (Start)

Constant term in formal Laurent series (Product_{i=1-n..n*(n-1)/2} (1+x^i)) - 1.

a(n) = (Sum_{i=0..n*(n-1)/2} A053632(n-1,i)*A000009(i))*2-1. (End)

Example

a(2) = 3 as there are three sets viz. {2), {1,3), {1,2,3}, the arithmetic mean of whose elements is 2.
a(3) = 9: the nine sets are {3}, {1, 5}, {2, 4}, {1, 2, 6}, {1, 3, 5}, {2, 3, 4}, {1, 2, 3, 6}, {1, 2, 4, 5}, {1, 2, 3, 4, 5}.

Maple

g := k->expand(mul(1+t*x^i, i=1..k)); A066571 := proc(n) local k; add(coeff(coeff(g(n*k), t, k), x, n*k), k=1..2*n-1); end;

Mathematica

g[k_] := Expand[Product[1 + t*x^i, {i, 1, k}]]; a[n_] := Sum[Coefficient[ Coefficient[g[n*k], t, k], x, n*k], {k, 1, 2*n - 1}]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 10}] (* Jean-François Alcover, Feb 10 2018, translated from Maple *)

Prog

(Haskell)
a066571 n = f [1..] 1 n 0 where
   f (k:ks) l nl x
     | y > nl  = 0
     | y < nl  = f ks (l + 1) (nl + n) y + f ks l nl x
     | otherwise = if y `mod` l == 0 then 1 else 0
     where y = x + k
-- Reinhard Zumkeller, Feb 13 2013

Crossrefs

Cf. A066572, A000217.

Cf. A072701, A164283.

Sequence in context: A073724 A151037 A371724 * A087648 A086616 A040027

Adjacent sequences: A066568 A066569 A066570 * A066572 A066573 A066574

Keyword

nonn,nice

Author

Amarnath Murthy, Dec 19 2001

Extensions

Corrected and extended by N. J. A. Sloane, Dec 19 2001

More terms from Naohiro Nomoto, Jun 19 2002

More terms from David Wasserman, Sep 10 2002

More terms from Martin Fuller, Sep 14 2023

Status

approved